Please help analytic geometry

A line with slope 6 bisects the area of a unit square with vertices (1,0), (0,0) , (1,1), and (0,1). What is the y-intercept of this line?

I tried putting one point where the line intersects the square as (y,1), and the other as (x,0), and the y intercept as (0,a). Then, I tried using the slope to figure out the variables. The only problem is that I don't have enough equations. Help?

asked by Knights
  1. I would say it hit the square at (a, 0) and at (b , 1)

    Area of trapezoid on left = (a+b)/2(1) = (a+b)/2

    Area of trapezoid on right = [(1-b)+(1-a)]/2
    = (2 - a - b)/2

    if those areas are equal then
    a + b = 2 - a - b
    or as we could see from the sketch
    a + b = 1

    Now the slope of 6
    b = a + (1/6)
    so
    a + a + 1/6 = 1
    2 a = 5/6
    a = 5/12
    then
    b = 5/12 + 1/6 = 7/12

    NOW you have a problem I am sure you can do
    find c in y = 6 x + c
    It goes through (5/12 , 0)
    0 = 5/2 + c
    so
    c = -5/2
    y = 6 x - 5/2
    for y axis intercept, x = 0
    y = -5/2

    posted by Damon
  2. You know the line is y=6(x-a) so it intercepts the x-axis at (0,a), and the line y=1 at x=(1+6a)/6 = 1/6 + a

    So, the left half of the figure is a trapezoid with height 1 and bases a and (1/6 + a)

    So, if that has area 1/2,
    1(a+a+1/6)/2 = 1/2
    a = 5/12

    so, y = 6(x-5/12)
    y = 6x - 5/2
    ---------------------------------
    Or, consider the point C=(1/2,1/2), the center of the square. A vertical line through C would cut the square in half.

    Since we want a line of slope 6, so y increases 6 for every increase in 1 by x. If the line passes through C, then if it leans to the right 1/12 at y=1 and to the left 1/12 at y=0, it will still cut the square in half. So, we have the line through (5/12,0) and (7/12,1).

    y = 6x - 5/2

    posted by Steve
  3. THANKS A LOT!

    posted by Knights

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